Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,13,16)(11,14,12,15)(17,18,22,20)(19,25,26,23)(21,28,30,24)(27,31,29,32)(35,41)(37,45)(39,42)(40,44);(18,20)(19,26)(24,28)(27,29)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(9,10)(11,14)(12,15)(13,16)(17,19,22,26)(18,23,20,25)(21,29,30,27)(24,32,28,31)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,20,22,18)(19,25,26,23)(21,28,30,24)(27,32,29,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=h,d^2=g,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4311857);